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Meaning of complex frequency

If we have a LC high pass filter the transfer function H(s) becomes:
~~![](https://physics.codidact.com/uploads/9ebwwQ7uz2nhUmVCM8WQow4u)~~
If we solve for s to find a pole of the transfer function we get:
~~![](https://physics.codidact.com/uploads/YFvV7yjmhDkMFSb1XatGiyxD)~~
~~In the case of a sinusoidal input signal = s = jω -> ω = 1/sqrt(LC)~~
But in a case of a signal which may not be sinusoidal s can be a complex number and the result of this is that the pole exists at a complex frequency.But what physical meaning does a complex frequency have?

If we have a LC high pass filter the transfer function H(s) becomes:
$$ H(s) = \cfrac{sL}{sL + \cfrac{1}{sC}} $$
If we solve for s to find a pole of the transfer function we get:
$$ s = j \cfrac{1}{\sqrt{LC}} $$
In the case of a sinusoidal input signal = $ s = j \omega \rightarrow \omega = \cfrac{1}{\sqrt{LC}} $
But in a case of a signal which may not be sinusoidal s can be a complex number and the result of this is that the pole exists at a complex frequency. But what physical meaning does a complex frequency have?

frequency electric-circuits